The case \(M=0\)  is trivial since the integrand in the definition of \(J(\widetilde{w};\theta )\)  is bounded and has compact support. For \(M\geqslant 1\) , we follow a similar argument to [1]}, by applying integration by parts and induction on \(M\) . We have
\(J(\widetilde{w};\theta ) = \int _{\mathbb {R}^4}\widetilde{w}(\mathbf {x})e(-\theta G(\mathbf {x})) \mathrm {d}\mathbf {x}\ll \max _{1 \leqslant j \leqslant 4}\left|\int _{\begin{array}{c}\mathbf {x}\in \mathbb {R}^4,\\|x_j| = \max _i|x_i|\end{array}}\widetilde{w}(\mathbf {x})e(-\theta G(\mathbf {x})) \mathrm {d}\mathbf {x}\right|.\) 
Without loss of generality, we may assume that \(j=1\) . We note that
\(e(-\theta G(\mathbf {x})) = \frac{1}{-4\pi i \epsilon _1\theta x_1}\frac{\partial }{\partial x_1}e(-\theta G(\mathbf {x})). \) 
We perform integration by parts with respect to \(x_1\)  to obtain
\(J(\widetilde{w};\theta ) \ll \left|\int _{\begin{array}{c}\mathbf {x}\in \operatorname{supp}{w}\\|x_1|= \max _i|x_i|\end{array}}\frac{\partial }{\partial x_1}\left(\frac{\widetilde{w}(\mathbf {x})}{4\pi i \theta x_1}\right)e(-\theta G(\mathbf {x}))\mathrm {d}\mathbf {x}\right|.\) 
Since \(|x_1| = \max _i|x_i|\) , we have \(|x_1|^{-1}= |\mathbf {x}|^{-1} \leqslant \eta ^{-1}\)  for any \(\mathbf {x}\in \operatorname{supp}{w}\) . Therefore by the assumptions on the size of \(\widetilde{w}\)  and its derivatives, we see that
\(\frac{\partial }{\partial x_1}\left(\frac{\widetilde{w}(\mathbf {x})}{4\pi i \theta x_1}\right) = \widetilde{w}^{(1)}(4\pi i\theta \eta ^2)^{-1}\) 
for some \(\widetilde{w}^{(1)}\in S\) . Hence
\(J(\widetilde{w};\theta )\ll |\eta ^2\theta |^{-1}|J(\widetilde{w}^{(1)};\theta )|.\) 
Proceeding by induction, we obtain \(J(\widetilde{w}(\mathbf {x});\theta )\ll _M |\eta ^2\theta |^{-M}|J(\widetilde{w}^{(M)};\theta )|\)  for some \(\widetilde{w}^{(M)}\in S\) , from which we deduce the required result by applying the trivial bound \(J(\widetilde{w}^{(M)};\theta ) \ll 1\) .
